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American Mathematical Monthly -March 2004

MONTHLY, March 2004

Yeuh-Gin Gung and Dr. Charles Y. Hu Award to T. Christine Stevens for Distinguished Service to Mathematics
by Robert E. Megginsonmeggin@msri.org

Pascal Matrices
by Alan Edelman and Gilbert Strangedelman@math.mit.edu, gs@math.mit.edu
Put the famous Pascal triangle into a matrix. It could go into a lower triangular L or its transpose L' or a symmetric matrix S. The entries are binomial coefficients (i/j) , and (j/i), and (i + j/i) for the symmetric matrix S.

The amazing thing is that L times L' equals S (known!). Since det L = 1 it follows that S has determinant 1. The matrices have other unexpected properties too, that give beautiful examples in teaching linear algebra. (Combinatorics too: The rows of the "hypercube matrix" L2 count corners and edges and faces in n-dimensional cubes.)

LL' = S is proved four ways, we don’t know which you will prefer:
1. By an identity for the binomial coefficients in these matrices.
2. By recognizing that each matrix counts paths on a directed graph.
Gluing the graphs for L and L' multiplies the matrices and gives S.
3. By induction: elimination on each matrix gives one size smaller.
* 4. By applying both sides to the column vector [ 1 xx2 x3 ... ]'.

This last way suggests a fifth proof, not complete, that would find a representation of all Mobius transformations by a matrix group including L and L' and S. Those matrices would represent the maps to x/(1-x) and 1+x and 1/(1-x). Then the composition of these transformations gives L L' = S. But the cube of the map from x to 1/(1-x) is the identity and we certainly doubt that S3 = I !

Interesting Dynamics and Inverse Limits in a Family of One-Dimensional Maps
by William T. Ingram and William S. Mahavieringram@umr.edu, wms@mathcs.emory.edu
In this article attention is restricted to the study of a simple family of piecewise linear unimodal maps from [0,1] onto [0,1]. The chaotic behavior of maps in this family was investigated by Susan Bassein in a paper in this MONTHLY in 1998. These same maps are used to give an elementary introduction to the study of inverse limits and to show that there are remarkable correlations between the chaotic nature of the maps in this family and the complexity of the inverse limits of the maps.

Fermat and the Quadrature of the Folium of Descartes
by Jaume Paradís, Josep PLa, and Pelegrí Viaderjaume.paradis@upf.edu, pla@mat.ub.es, pelegri.viader@upf.edu
Do you think you know everything about elementary integration? Try to find the area inside the loop that the folium of Descartes, the curve described by the cubic x3 + y3 = xy, bounds in the first quadrant. You will undoubtedly succeed, but only after some effort and with quite heavy artillery at your disposal: the integral calculus. Perhaps a generalization of the folium proposed in 1917 will be more of a challenge: x2q+1 + y2q+1 = (2q+1)xq yq. But, what can you do about the generalization we propose in this article: x2q+1 + y2q+1= xy? Fermat, with a method of his own based on elementary means was able, in the parlance of his time, to square the loop of the folium of DescartesÂ—quite an achievement in the mid-seventeenth century. We will see how Fermat’s method works and how can it be applied to attack the more formidable generalizations just indicated.

The Spiral of Theodorus
by Detlef Gronaugronau@uni-graz.at
The discrete spiral of Theodorus can be represented in the complex plane by z0 = 1 and zn + 1, (1 + i/√n +)zn n = 0,1,2,Â… (see Philip J. Davis, Spirals from Theodorus to Chaos, A. K. Peters, 1993). Davis proposes the "Theodorus function" T, defined by an infinite product, as an interpolating function of the discrete spiral of Theodorus. The function T satisfies a first order difference equation. Davis asked for a (geometrical) characterization of the Theodurus function among the various solutions of this difference equation. In our paper we give some characterizations of this function. We give a new formula for the Theodorus function that converges faster than the infinite product of Davis. We also provide pictures of some different spirals and spiral-like curves.

Putting Fractions in Their Place
by Leslie Blackwell Galenleslie@integretechpub.com
Authors, students, and editors should learn to recognize that mathematics has a typography all its own, and should feel comfortable reading, writing, and typesetting mathematical expressions. This article deals with fractions. Fractions are explained as typographical entities, rather than mathematical functions. The four kinds of fractions—case, special, shilling, and built-up—are discussed, as well as where and how they should be used. Examples of misuse, as well as tips for proper use and setting of different sorts of fractions, are shown with examples and discussion.

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